AQ will be equal to radius, that's what our teacher told.
She said, line segment from A on hypotenuse OP at its midpoint will be equal to the radius.
But, what's the justification? What reason should I write?

Take OI=x. IP is this, (2r-x). Using pythagoras' theorem, find AI from triangle AIP(AP=r*sqrt3) and then from triangle AOI. Equate those two. You will get x=r/2. After that, prove the congruency of AOI and AQI. You will get AQ=r. Do I need to show you the steps?